{
  "id": "0904.3350",
  "version": "v3",
  "published": "2009-04-21T21:35:19.000Z",
  "updated": "2012-03-28T20:47:43.000Z",
  "title": "Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory",
  "authors": [
    "Kiumars Kaveh",
    "A. G. Khovanskii"
  ],
  "comment": "39 pages. Revised in several places and the title slightly modified. Final version, to appear in Annals of Mathematics",
  "categories": [
    "math.AG",
    "math.AC",
    "math.CO"
  ],
  "abstract": "Generalizing the notion of Newton polytope, we define the Newton-Okounkov body, respectively, for semigroups of integral points, graded algebras, and linear series on varieties. We prove that any semigroup in the lattice Z^n is asymptotically approximated by the semigroup of all the points in a sublattice and lying in a convex cone. Applying this we obtain several results: we show that for a large class of graded algebras, the Hilbert functions have polynomial growth and their growth coefficients satisfy a Brunn-Minkowski type inequality. We prove analogues of Fujita approximation theorem for semigroups of integral points and graded algebras, which imply a generalization of this theorem for arbitrary linear series. Applications to intersection theory include a far-reaching generalization of the Kushnirenko theorem (from Newton polytope theory) and a new version of the Hodge inequality. We also give elementary proofs of the Alexandrov-Fenchel inequality in convex geometry and its analogue in algebraic geometry.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-03-28T20:47:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C20",
      "13D40",
      "52A39"
    ],
    "keywords": [
      "graded algebras",
      "integral points",
      "intersection theory",
      "newton-okounkov body",
      "newton polytope theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0904.3350K"
    }
  }
}